LBO is buying/acquisition of a company using debt instruments issued either to the seller or third party. MBO is purchase/acquisition of a company by the management team and a MBO can also be a LBO.
An MBO is a subset of an LBO. The LBO is the use of an extremely high amount of financial leverage (debt) by a firm (or group of investors) to acquire another (target) firm. The MBO is a type of LBO where part of the acquiring investors (a firm and/or invesotrs) includes the senior managers (officers) of the to be acquired (target) firm. There is an appearance, at the least, of conflict of interest between the managers (officers) of the target firm and the shareholders of the target firm in an MBO that somehow is dealt with by the investment bankers and lawyers.